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GENERATING INFINITE SYMMETRIC GROUPS

Published online by Cambridge University Press:  31 May 2006

GEORGE M. BERGMAN
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA gbergman@math.berkeley.edu
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Abstract

Let $S={\mathop{\rm Sym}(\Omega)$ be the group of all permutations of an infinite set $\Omega$. Extending an argument of Macpherson and Neumann, it is shown that if $U$ is a generating set for $S$ as a group, then there exists a positive integer $n$ such that every element of $S$ may be written as a group word of length at most $n$ in the elements of $U$. Likewise, if $U$ is a generating set for $S$ as a monoid, then there exists a positive integer $n$ such that every element of $S$ may be written as a monoid word of length at most $n$ in the elements of $U$. Some related questions and recent results are noted, and a brief proof is given of a result of Ore's on commutators, which is used in the proof of the above result.

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Copyright
© The London Mathematical Society 2006

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